关于与APN映射相关的不同概念
On different notions related to APN mappings
- Univerza na Primorskem - Fakulteta za matematiko, naravoslovje in informacijske tehnologije（滨海大学数学、自然科学与信息技术学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究了与APN映射相关的多种概念，包括k-breaking、强非正规性和和自由性，引入k-强breaking并刻画APN函数子类，通过特征变换解决开放问题并探索平衡性质。
AI中文摘要：
一个APN映射 $F:\mathbb{F}_{2^n}\to \mathbb{F}_{2^n}$ 是一个多项式，其特征是在2-平坦集上具有非消失性质。在本工作中，我们分析了与此性质密切相关的概念。为了理解 $\mathbb{F}_{2^n}$ 的哪些 $k$-平坦集在 $F$ 作用下仍然是平坦集，我们研究了 $k$-breaking 性质。函数 $x^{-1}$ 过去曾在此背景下被研究——我们将此研究扩展到一般映射，并刻画了APN函数的2-breaking性质。最近，APN性质的两个推广被引入：$k$-强非正规性和$k$阶和自由性。和自由性将APN函数的非消失性质推广到更高维平坦集。我们深入观察了breaking性质、强非正规性和和自由性之间的关系。我们证明了一个3阶和自由APN函数必定是3-breaking的。我们引入了第四个概念，称为$k$-强breaking，它蕴含breaking性质。我们为这两个概念推导出若干结构结果，并给出了APN函数的一个子类在2-强breaking性质下的刻画。我们通过一种自然的特征变换提出了非消失性质的不同视角，该变换与$F$的分量的平方和指标密切相关。我们推导了$F$的平方和指标总和的精确值。利用这种方法，我们为IEEE Trans. Inf. Theory 52(9): 4160-4170, 2006中的开放问题4提供了一个简单答案。此外，它使我们能够探索多项式的平衡性质，其中之一在奇数$n$下刻画了逐分量的APN性，并为任意维度提供了自然推广。我们证明了Dillon的APN置换和Gold函数满足一个相关性质，称为$k$-平衡，该性质在我们的框架下呈现。
英文摘要：
An APN mapping $F:\mathbb{F}_{2^n}\to \mathbb{F}_{2^n}$ is a polynomial characterized by the non-vanishing property on 2-flats. In this work, we analyze notions that are closely related to this property. To understand which $k$-flats of $\mathbb{F}_{2^n}$ remain flats under $F$, we study the $k$-breaking. The function $x^{-1}$ has been studied in the past in this context---we extend this study to general mappings and characterize the 2-breaking of APN functions. Recently, two generalizations of the APN property have been introduced: $k$-strongly non-normality and $k$-th-order sum-freedom. Sum-freedom generalizes the non-vanishing property of APN functions to higher dimensional flats. We provide in-depth observations of the relations between the breaking property, strongly non-normality and sum-freedom. We introduce a fourth concept called $k$-strongly breaking, which implies the breaking property. We derive several structural results for both notions and give a characterization of a subclass of APN functions in terms of the 2-strongly breaking property. We propose a different perspective of the non-vanishing property via a natural character transformation, which is closely related to the sum-of-square indicator of the components of $F$. We derive a precise value for the total sum of the sum-of-square indicators of $F$. With this approach, we provide a simple answer to Open Problem 4 in IEEE Trans. Inf. Theory 52(9): 4160-4170, 2006. Moreover, it allows us to explore balancedness properties of polynomials, one of which characterizes component-wise APNness, for odd $n$, and provides a natural extension to any dimension. We show that Dillon's APN permutation and the Gold functions satisfy a related property, termed $k$-balanced, which is presented under our framework.
